INTRODUCTION AND PLAN OF THE CHAPTER

Greek proofs prove general results. Whatever its object, a Greek proof is a particular, an event occurring on a given papyrus or in a given oral communication. The generality of Greek mathematics should therefore be considered surprising. And this is made even more surprising following the argument of chapter 1 above, that Greek mathematical proofs are about specific objects in specific diagrams. The following chapter, then, tries to account for a surprising feature, a paradox.

The nature of this chapter must be different from that of the preceding one. I have explained necessity in terms of atomic necessity-producing elements, which are then combined in necessity-preserving ways. But there are no atoms of generality, there are no ‘elements’ in the proof which carry that proof's generality. Generality exists only on a more global plane. This global nature has important implications. That which exists only on the abstract level of structures cannot be present to the mind in the immediate way in which the necessity of starting-points (chapter 5, section 1), say, is present to the mind. The discussion of necessity focused on a purely cognitive level. It was psychological rather than logical. This chapter will have to be more logical.

This is not to say that the chapter should be judged by its success in reconstructing a lost logical theory, once fully developed by Greek mathematicians. The theory which explicates and validates a practice may be only partially understood by those who follow that practice.